Proving Trace Independence in Linear Algebra Homework

In summary, the trace of a matrix is independent of the orthonormal basis chosen, as long as the matrix is diagonalizable and the eigenvectors are used as the basis vectors. This is because the trace is the sum of the eigenvalues, which remain constant regardless of the chosen basis. To show this, one must demonstrate the relationship between the original matrix and the same matrix written in a different basis.
  • #1
zheng89120
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Homework Statement



let:
Trace(A) = Ʃ(i=1..n) (ei|A|ei)

Show that trace is independent of the orthonormal basis chosen.

Homework Equations



linear algebra

The Attempt at a Solution



trace is related to the eigenvalues, which are constant?
 
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  • #2
zheng89120 said:

Homework Statement



let:
Trace(A) = Ʃ(i=1..n) (ei|A|ei)

Show that trace is independent of the orthonormal basis chosen.

Homework Equations



linear algebra

The Attempt at a Solution



trace is related to the eigenvalues, which are constant?

IF A a diagonalizable and IF the ei happen to be eigenvectors, then sure, it's easy to see the trace is the sum of the eigenvalues. But how does that help you show it's basis independent? How do you write the relation between A and A written in a different basis?
 

FAQ: Proving Trace Independence in Linear Algebra Homework

What is linear algebra and why is it important?

Linear algebra is a branch of mathematics that deals with linear equations, vectors, and matrices. It is important because it provides powerful tools for solving problems in various fields such as physics, engineering, computer science, and economics.

What are the basic concepts in linear algebra?

The basic concepts in linear algebra include vectors, matrices, linear transformations, and systems of linear equations. These concepts are used to represent and solve real-world problems using mathematical equations and operations.

What are the applications of linear algebra?

Linear algebra has many applications in different fields such as computer graphics, machine learning, data analysis, and cryptography. It is also used in solving problems related to optimization, statistics, and differential equations.

What is the difference between a vector and a matrix?

A vector is a quantity that has both magnitude and direction, represented by an ordered list of numbers. A matrix, on the other hand, is a rectangular array of numbers arranged in rows and columns. Vectors are often used to represent points in space, while matrices are used to represent transformations and systems of equations.

How can linear algebra be used to solve real-world problems?

Linear algebra provides a systematic and efficient way to solve problems by representing them in the form of equations and using algebraic operations to manipulate these equations. It also helps in visualizing and understanding complex data and relationships between variables, leading to better problem-solving techniques.

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